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*Lectures on Quantum Mechanics*, Weinberg makes the following claim about solutions to the Schrodinger equation for a central potential: Suppose [itex]\psi(\mathbf{x},t)[/itex] is an eigenfunction of [itex]H[/itex], [itex]\mathbf{L}^2[/itex], and [itex]L_z[/itex]. According to Weinberg, "since [itex]\mathbf{L}^2[/itex] acts only on angles, such a wavefunction must be proportional to a function only of angles, with a coefficient of proportionality [itex]R[/itex] that can depend only on [itex]r[/itex]. That is, for all r,

[tex]\psi(\mathbf{x})=R(r)Y(\theta,\phi)."[/tex]

He does not elaborate further on this, in my view, non-trivial statement. Can someone here provide a proof of his claim?